Interior Gradient Estimates And Liouville Type Result For Mean Curvature Type Equations

In this paper,we mainly study the interior gradient estimates of mean curvature type equation and mean curvature flow equation with parameter .And it is concluded that the singular minimal surface equation is proved to achieve the expected Liouville type results by using the interior gradient estimates.In chapter one,we briefly introduce the research background for mean curvature equation and the singular minimal surface equation and the process of proposing this problem in this paper.In chapter two,we mainly introduce the preparation knowledge of the interior gradient estimates of mean curvature equation and the singular minimal surface equation.Firstly,we introduce the research background and simple derivation process of the mean curvature equation,and then give the maximum principle of the heat conduction equation and the general parabolic equation?elliptic equation,finally give the theorem of gradient interior estimates.In chapter three,we study the interior gradient estimates for mean curvature type equations and mean curvature flow equations with parameter .Firstly,we give a proof of the interior gradient estimate for mean curvature type equation,and then prove that the mean curvature flow equation with parameter can also obtain similar interior gradient estimate.In chapter four,we study the expected Liouville type result for the singular minimal surface equation.First,we obtain the interior gradient estimates for the singular minimal surface equation,and then give the Liouville type result for the singular minimal surface equation.